Evaluate: $\int_{0}^{\pi / 4} \sin 2 x \sin 3 x\, d x$.

CBSE Class 12 · 2006 · Definite IntegralsLong Answer

Evaluate:

\int_{0}^{π /4} sin 2 x sin 3 x d x

Answer: $\frac{3}{5 2}$

Convert the product to a sum using

sin A sin B = \frac{1}{2} [cos (A - B) - cos (A + B)]

sin 2 x sin 3 x = \frac{1}{2} (cos x - cos 5 x)

I = \frac{1}{2} \int_{0}^{π /4} (cos x - cos 5 x) d x = \frac{1}{2} [sin x - \frac{sin 5 x}{5}]_{0}^{π /4}

Now

sin \frac{π}{4} = \frac{1}{2}

and

sin \frac{5 π}{4} = - \frac{1}{2}

= \frac{1}{2} [\frac{1}{2} + \frac{1}{5 2}] = \frac{1}{2} \cdot \frac{6}{5 2} = \frac{3}{5 2}

CBSE Class 12 Board · 2006 · Subjective